In a climate application we are interested in computing the rate function
\(I(a)\) . However, the direct estimation of the rate function is not
usually the best way to proceed [1]}, as it is difficult
to provide quantitative arguments to justify the convergence of estimators
of probability density functions. A more precise way of proceeding
is to compute the scaled cumulant generating function first, and then
to compute the rate function as its Legendre–Fenchel transform.
Following this approach, the first step is to compute an estimate
of the generating function approximating the expectation value with
the average over the \(N_{b}\)  blocks in the limit of large \(N_{b}\)
\(\hat{G}(k,\tau _{b},N_{b})=\frac{1}{N_{b}}\sum _{j=1}^{N_{b}}e^{kA_{\tau _{b}}^{j}}.\) 
